Integration of Gravitational Torques in Cerebellar Pathways Allows for the Dynamic Inverse Computation of Vertical Pointing Movements of a Robot Arm

Several authors suggested that gravitational forces are centrally represented in the brain for planning, control and sensorimotor predictions of movements. Furthermore, some studies proposed that the cerebellum computes the inverse dynamics (internal inverse model) whereas others suggested that it computes sensorimotor predictions (internal forward model).This study proposes a model of cerebellar pathways deduced from both biological and physical constraints. The model learns the dynamic inverse computation of the effect of gravitational torques from its sensorimotor predictions without calculating an explicit inverse computation. By using supervised learning, this model learns to control an anthropomorphic robot arm actuated by two antagonists McKibben artificial muscles. This was achieved by using internal parallel feedback loops containing neural networks which anticipate the sensorimotor consequences of the neural commands. The artificial neural networks architecture was similar to the large-scale connectivity of the cerebellar cortex. Movements in the sagittal plane were performed during three sessions combining different initial positions, amplitudes and directions of movements to vary the effects of the gravitational torques applied to the robotic arm. The results show that this model acquired an internal representation of the gravitational effects during vertical arm pointing movements.This is consistent with the proposal that the cerebellar cortex contains an internal representation of gravitational torques which is encoded through a learning process. Furthermore, this model suggests that the cerebellum performs the inverse dynamics computation based on sensorimotor predictions. This highlights the importance of sensorimotor predictions of gravitational torques acting on upper limb movements performed in the gravitational field

Contrôle moteur par le cervelet et interface Cerveau-Machine pour commander un doigt robotique

This thesis focuses on motor control modelling in primate, using two specific approaches: -First, following a methodology based on mathematical formalization for the preparation by the Brain, of control signals driving a voluntary movement directed toward a target. The methodology used in this study consists in identifying the functional constraints and in deriving a processing circuit for motor signals, compatible with the anatomical organization of cerebellar pathways, and allowing a hierarchical optimization under the constraints of timeliness and energy-saving expenditure. This approach was applied to the control of a robotic arm with 2 DOF actuated by McKibben muscles and also to the modelling of the primate oculomotor system. -Secondly, following a coding approach. We present here the design and development of an asynchronous brain-machine interface that decodes the data recorded in the Macaque brain to control a robotic finger.Cette thèse porte sur la modélisation de la commande motrice chez le Primate suivant deux approches : - Tout d'abord, en suivant une formalisation mathématique de la préparation par le Cerveau des signaux de commande d'un mouvement volontaire dirigé vers une cible. La méthode utilisée dans cette étude a été de recenser les contraintes fonctionnelles et d'en déduire un circuit de traitement des signaux moteurs, compatible avec l'organisation anatomique des voies cérébelleuses. Ce circuit a permis une optimisation hiérarchisée, sous les contraintes de rapidité d'exécution et d'économie de la dépense énergétique. Cette approche a été appliquée à la commande d'un bras robotique à 2 d.d.l mû par des muscles de McKibben, et à la modélisation du système oculomoteur du Primate. - Ensuite, en suivant une approche par codage. Nous présentons ici la conception et la mise au point d'une Interface Cerveau-Machine asynchrone qui décode les données cérébrales enregistrées chez le Macaque afin de contrôler un doigt robotique

Modélisation des voies de la commande saccadique

A model of the saccadic pathways is proposed

Learning curves for robotic experiment.

<p>(A) Three learning curves obtained for the three mass conditions M0, M1 and M2. (B) Typical desired and performed arm displacements from a horizontal position (i.e. 0°) and with an amplitude of 25°. DaD: Desired angular Displacement. PaD: Performed angular Displacement.</p

Pointing errors for simulation 2.

<p>Average RMSE_D (D) and RMSE_S (S) for each mass condition for the session I (SI), II (SII) and III (SIII). Training: Training set. Iep (inter- and extrapolated positions): test set. M_i_T (0≤i≤5): masses used during the training set. M_i_Iep (0≤i≤5): masses used during the test set. Average Iep: RMSE values for the test set averaged across SI, SII and SIII. Aver M0–5_T, Aver M0–5_T; Aver M0–5_Iep: RMSE_D and RMSE_S values for the training (_T) and test set (_Iep) averaged across the different mass conditions, respectively.</p

Schematic explanation of the setup and the three simulated tasks.

<p>Representation of the simulated arm and its two antagonist muscles. F1: force developed by muscle 1. F2: force developed by muscle 2. R: radius of the sprocket. Black arrow: gravitational torque (GT) exerted on the segment. First row: Session I. Initial position (green circle) at 0°; Required movements: upward (red arrow), downward (blue arrow). Traces to the right: gravitational torque over time as a function of target movement amplitude and direction (upward: red, downward: blue). For all movements of session I, the gravitational torque varies monotonically and is independent of movement direction. Second row: Session II. Initial positions at 20° and −20°. The gravitational torque varies monotonically but depends on movement direction. Third row: Session III. Initial positions at 20° and −20°. For each movement (amplitudes: 40° and 60°) the gravitational torque varies non-monotonically.</p

Main principles of the cerebellar architecture of the model.

<p>For the sake of clarity, only the command circuit for one muscle is illustrated. (A) Structure of a command circuit accounting for the physical constraints. θ^D: desired movement; θ^P: performed movement; u: neural command. H: direct function incorporating all biophysical features of the limb. H⁻¹: internal inverse model of the direct function. (B) Control scheme used to compute an approximate inverse function. The two summing elements (positive/negative inputs) represent the cerebellar nuclei (CN) and the red nucleus (RN). H*: internal forward model of the direct function H. P and Q represent the signals originating from the cerebellar cortex (CC) and CN, respectively. (C) Direct functions Π* (representing the mechanical constraints, e.g. gravity, inertia) and μ (representing the muscle features) in the external world, and their counterparts in the CNS labeled Π* and μ*. These two internal forward models (Π* and μ*) are embedded through two internal feedback loops placed in series to calculate their approximate inverse, i.e., Π⁻¹and μ⁻¹. The direct pathways convey signals of desired position θ^D and forces F^D. In the indirect pathways, the negative output of the elements computing Π* and μ* are comparable to the inhibitory projections of the Purkinje cells of the cerebellar cortex to the neurones of the cerebellar nuclei. The P signals are comparable to simple spike activities of Purkinje cells. Dashed lines represent the climbing fibers. SC: spinal cord. Lower scheme: artificial neural networks simulating the CC. Granular, Golgi and Purkinje cells (respectively 8, 1 and 1 for each predictor) are modelled by formal neurones. s and s⁻¹ represent respectively the derivative and integration in the Laplace domain. α represents multiplicative higher orders of the position. The adaptive elements and connections are represented in grey, fixed elements in black. For the sake of the clarity only the first neural network is represented.</p

Performance of the model for simulation 2.

<p>Pointing error as a function of weight (mass condition M0_Iep, M1_Iep, M2_Iep, M3_Iep, M4_Iep and M5_Iep) and movement amplitude (and initial position). First column: dynamic error (RMSE_D). Second column: static error (RMSE_S). Each black dot represents an error measure (dynamic or static) obtained for a given experimental condition. IP: initial position.</p

Comparison between simulated and robot movements.

<p>Distribution of the RMSE_D (left column) and RMSE_S (right column) for the three sessions for the simulation 1 (A) and the robotic experiment (B). Both type of error are represented as a function of movement amplitudes during the session I (i.e. intra- and extrapolated positions) and of initial positions and movement amplitudes during respectively the session II and III.</p

Pointing errors for simulation 1.

<p>Average RMSE_D (D) and RMSE_S (S) for each mass condition for the session I (SI), II (SII) and III (SIII). Training: Training set. Iep (inter- and extrapolated positions): test set. Average Iep: RMSE values for the test set averaged across SI, SII and SIII. Average M0–2: RMSE_D and RMSE_S values for both the training and test set averaged across the different mass conditions (M0, M1, M2) for each session (SI, SII, SIII).</p